I am having a bit of trouble evaluating $$\sum_{k=1}^3{
\rm Res}\left(\frac{\log(z)}{z^3+8};z_k\right)$$ where $z_1=2e^{i\pi}$, $z_2=2e^{i\pi/3}$ and $z_3=2e^{i5\pi/3}$. I know that each $z_k$ is a ...

$\int_0^\infty$ $\frac{1}{1+x}$$\frac{dx}{\sqrt{x}}$
Part (a) asks to compute the integral by means of the residue at x = -1. I have done this just now, and the answer is $\pi$.
Part (b) asks, "can ...

I have to show summability, then compute the following integral:
$$\int\limits_{-\infty}^{+\infty} \frac{\sin(\pi\,x)}{\prod_{k = - n}^n (x - k)}\,dx = \frac{(-4)^n}{(2\,n)!}\,\pi $$
for every $n\in ...

I want to solve the integral attached below by means of residue theorem. I tried the common integration ways and seeked references(e.g, Rjadov, et. al).
Finally, I decided to solve this integral by ...

So the question is as follows: Use the Residue Theorem to calculate $$\int_0^{2\pi} \frac{1}{2\pi\cos^{2n}(\theta)} d\theta \quad\quad n=1,2,3,\dots.$$
Now I believe the first step would be to use the ...

Say I have this integral: $$\oint_\gamma f(z)\,{\rm d}z,$$and $f$ has a pole on $\gamma$. I understand that we "cut around" the pole with an arc of radius $\epsilon$ and then make $\epsilon \to 0$. ...

I have to solve the next integral:
$$\int_{-\infty}^{\infty} e^{ibx}(e^{ia/x}-1)dx$$ where $a,b$ are real parameters.
I can use Jordan´s Theorem to show that as $f(z)=e^{ibz}g(z)$ where $g(z)=(e^{ ia ...

I'm trying to calculate the residua of the following complex function but am encountering problems trying to determine its poles:
$$f(z)=\frac{\sin(z)}{z^4}$$
Expanding the denominator shows that we ...

I need to find residue of function $f(z) = \frac{z}{1-\cos(z)}$ at $z=2\pi k$, where $k\in \Bbb Z$.
I know residue at $z=0$ from here.
I got a hint that need to substitute $z=\hat z+2\pi k$, so $\hat ...

I am trying to use the residue theorem to evaluate $$I=\int_0 ^{\infty}\frac{dx}{x^{1/3}(1+x)}$$ I'll explain my difficulty in finding a contour, then I explain my difficulty in finding a new contour ...